Finite Group Databasebeta

← C3×#10591565 #13158569

Subgroups — C3×#10591565 #13158569

Order 1728.

696 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3755C211yesmaximal, characteristic
2C32×#944C211yesmaximal, characteristic, complemented
3C3×#63894C211yesmaximal, characteristic
4C2×C3×#3717C211yesmaximal, characteristic
5C32×#835C211yesmaximal, characteristic, complemented
6C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7C3×#63565C211yesmaximal, characteristic, complemented
8C3×#5083—31nomaximal
9C3×#4818—31nomaximal
10#10591565C311yesmaximal, characteristic, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C3×#624—21no—
15C3×#3755—22no—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×Dic3⋊C4—21no—
19C32×Dic3⋊C4—21no—
20C32×C4×Dic3C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×#3722—21no—
23C3×#3717—22no—
24C32×C4⋊Q8—31no—
25C2×#3755C2×C311yescharacteristic
26C3×#944—21no—
27C3×#944D311yescharacteristic, complemented
28C3×#944C2×C311yescharacteristic, complemented
29C2×C3×#215—31no—
30C2×C3×#215—31no—
31C2×C3×#215—31no—
32C2×C3×#215—61no—
33C2×C3×#204—31no—
34C3×#893—31no—
35C3×C4×D12—31no—
36#63894C2×C311yescharacteristic
37C2×#3717C2×C311yescharacteristic
38C32×#103—31no—
39C3×#835—21no—
40C3×#835—31no—
41C3×#835D311yescharacteristic, complemented
42C3×#835C2×C311yescharacteristic, complemented
43C3×Dic3⋊C8—31no—
44C3×#810—31no—
45C3×C4⋊(C3⋊C8)—31no—
46Dic3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
47#63565C2×C311yescharacteristic, complemented
48C22×C3×C3⋊D3—21no—
49C33×Q8—21no—
50C33×Q8—21no—
51C2×C33×C4—21no—
52C2×C33×C4C2311yescharacteristic
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)D412yes—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D411yescharacteristic
58C2×C32×Dic3D411yescharacteristic
59C32×Dic6—21no—
60C32×Dic6—21no—
61C3×#624—42no—
62C3×C3⋊(C3⋊C8)—21no—
63C32×C3⋊C8—21no—
64C3×#461—91no—
65#5083—31no—
66#4818—31no—
67C2×C32×Q8—21no—
68C2×C32×Q8—21no—
69C2×C32×Q8—21no—
70C2×C32×Q8—31no—
71C2×C32×Q8—41no—
72C2×C32×Q8C2×D311yescharacteristic
73C2×C32×Q8C2×D311yescharacteristic
74C2×C32×Q8C22×C311yescharacteristic
75C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
76C2×C3×D12—31no—
77C2×C3×D12—31no—
78C2×C3×D12—61no—
79C2×C3×C4×D3—61no—
80C2×C3×C4×D3—61no—
81C2×C3×Dic6—21no—
82C2×C3×Dic6C2×D311yescharacteristic
83C2×C3×Dic6C22×C311yescharacteristic
84C2×#624—21no—
85C2×C32×C8—31no—
86C32×C4⋊C4—31no—
87C32×C4⋊C4—61no—
88C32×C4⋊C4—61no—
89C32×C42—31no—
90#3755—22no—
91C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
92C3×#215—62no—
93C3×#215—62no—
94C3×#215—62no—
95C3×#215—64no—
96C3×(C2×D3)⋊C4—61no—
97C3×(C2×D3)⋊C4—61no—
98C3×C4⋊Dic3—21no—
99C3×C4⋊Dic3C2×D311yescharacteristic
100C3×C4⋊Dic3C22×C311yescharacteristic
101C3×Dic3⋊C4—22no—
102C3×Dic3⋊C4—22no—
103C3×Dic3⋊C4—21no—
104C3×Dic3⋊C4—21no—
105C3×Dic3⋊C4—21no—
106C3×Dic3⋊C4—21no—
107C3×C4×Dic3—21no—
108C3×C4×Dic3C2×D311yescharacteristic
109C3×C4×Dic3C22×C311yescharacteristic
110C2×C3×C3⋊C8—21no—
111C2×C3×C3⋊C8—31no—
112C2×C3×C3⋊C8—31no—
113C2×C3×C3⋊C8—61no—
114C2×C3×C3⋊C8C2×D311yescharacteristic
115C2×C3×C3⋊C8C22×C311yescharacteristic
116C3×#204—62no—
117#3722—21no—
118#3717—22no—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—22no—
121C2×C3×C3⋊D3—41no—
122C2×C3×C3⋊D3—41no—
123C33×C4—41no—
124C33×C4C2×D411yescharacteristic
125C33×C4#4111yescharacteristic
126C32×Dic3—21no—
127C32×Dic3—41no—
128C32×Dic3—41no—
129C32×Dic3SD1612yescomplemented
130C2×C3×SD16—91no—
131C2×C3×SD16—91no—
132C3×C4⋊Q8—31no—
133C3×C4⋊Q8—31no—
134C3×C4⋊Q8—61no—
135C3×C4×D4—91no—
136#944C3×D311yescharacteristic, complemented
137C2×#215—31no—
138C2×#215—31no—
139C2×#215—31no—
140C2×#215—61no—
141C2×#204—31no—
142#893—31no—
143C4×D12—31no—
144C3×C4⋊C8—91no—
145C3×#103—31no—
146C3×#103—31no—
147C3×#103—61no—
148C3×#103—91no—
149#835—31no—
150#835C3×D311yescharacteristic, complemented
151Dic3⋊C8—31no—
152#810—31no—
153C4⋊(C3⋊C8)—31no—
154C22×C3⋊D3—21no—
155C22×C3×D3—61no—
156C22×C3×D3—61no—
157C22×C3×D3—121no—
158C32×Q8—21no—
159C32×Q8—21no—
160C32×Q8—22no—
161C32×Q8—22no—
162C32×Q8—21no—
163C32×Q8—21no—
164C32×Q8—21no—
165C32×Q8—21no—
166C32×Q8—41no—
167C32×Q8—41no—
168C32×Q8—41no—
169C32×Q8—41no—
170C32×Q8—42no—
171C32×Q8—42no—
172C32×Q8—61no—
173C32×Q8—61no—
174C2×C32×C4—21no—
175C2×C32×C4—22no—
176C2×C32×C4—21no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4—31no—
182C2×C32×C4—31no—
183C2×C32×C4—41no—
184C2×C32×C4—41no—
185C2×C32×C4—41no—
186C2×C32×C4—42no—
187C2×C32×C4—61no—
188C2×C32×C4—61no—
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C23×C311yescharacteristic
192C4⋊(C3⋊D3)—21no—
193C4⋊(C3⋊D3)C3×D412yes—
194C2×C3×Dic3—22no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—21no—
197C2×C3×Dic3—21no—
198C2×C3×Dic3—21no—
199C2×C3×Dic3—21no—
200C2×C3×Dic3—21no—
201C2×C3×Dic3—41no—
202C2×C3×Dic3#6711yescharacteristic
203C2×C3×Dic3#6711yescharacteristic
204C2×C3×Dic3C3×D411yescharacteristic
205C2×C3×Dic3C3×D411yescharacteristic
206C3×D12—32no—
207C3×D12—32no—
208C3×D12—62no—
209C3×D12—61no—
210C3×D12—61no—
211C3×D12—121no—
212C3×C4×D3—122no—
213C3×C4×D3—122no—
214C3×Dic6—21no—
215C3×Dic6—21no—
216C3×Dic6—21no—
217C3×Dic6—21no—
218C3×Dic6—41no—
219C3×Dic6—41no—
220#624—42no—
221C32×C8—61no—
222C3⋊(C3⋊C8)—21no—
223C3×C3⋊C8—21no—
224C3×C3⋊C8—21no—
225C3×C3⋊C8—41no—
226C3×C3⋊C8—62no—
227C3×C3⋊C8—61no—
228C3×C3⋊C8—61no—
229#461—91noSylow
230C2×C33#12111yescharacteristic
231C2×C33C2×SD1611yescharacteristic
232C2×C33#13711yescharacteristic
233C3×C3⋊D3—42no—
234C2×C3×Q8—21no—
235C2×C3×Q8—21no—
236C2×C3×Q8—21no—
237C2×C3×Q8—31no—
238C2×C3×Q8—31no—
239C2×C3×Q8—41no—
240C2×C3×Q8—61no—
241C2×C3×Q8D3211yescharacteristic
242C2×C3×Q8C2×C3×D311yescharacteristic
243C2×C3×Q8C2×C3×D311yescharacteristic
244C2×C3×D4—91no—
245C22×C3×C4—181no—
246C2×D12—31no—
247C2×D12—31no—
248C2×D12—61no—
249C2×C4×D3—61no—
250C2×C4×D3—61no—
251C2×Dic6C2×C3×D311yescharacteristic
252C3×SD16—182no—
253C3×SD16—182no—
254C2×C3×C8—31no—
255C2×C3×C8—31no—
256C2×C3×C8—61no—
257C2×C3×C8—91no—
258C3×C4⋊C4—31no—
259C3×C4⋊C4—31no—
260C3×C4⋊C4—61no—
261C3×C4⋊C4—61no—
262C3×C4⋊C4—61no—
263C3×C4⋊C4—61no—
264C3×C4⋊C4—61no—
265C3×C4⋊C4—62no—
266C3×C4⋊C4—62no—
267C3×C22⋊C4—181no—
268C3×C42—31no—
269C3×C42—31no—
270C3×C42—61no—
271#215—62no—
272#215—62no—
273#215—62no—
274#215—64no—
275(C2×D3)⋊C4—61no—
276(C2×D3)⋊C4—61no—
277C4⋊Dic3C2×C3×D311yescharacteristic
278Dic3⋊C4—21no—
279Dic3⋊C4—21no—
280C4×Dic3C2×C3×D311yescharacteristic
281C2×C3⋊C8—31no—
282C2×C3⋊C8—31no—
283C2×C3⋊C8—61no—
284C2×C3⋊C8C2×C3×D311yescharacteristic
285#204—62no—
286C22×C32—21no—
287C22×C32—21no—
288C22×C32—21no—
289C22×C32—41no—
290C22×C32D3×D411yescharacteristic
291C22×C32C2×#6711yescharacteristic
292C22×C32C2×C3×D411yescharacteristic
293C2×C3⋊D3—22no—
294C2×C3⋊D3—41no—
295C2×C3⋊D3—41no—
296C2×C3×D3—62no—
297C2×C3×D3—62no—
298C2×C3×D3—122no—
299C2×C3×D3—121no—
300C2×C3×D3—121no—
301C2×C3×D3—121no—
302C2×C3×D3—121no—
303C2×C3×D3—241no—
304C2×C3×D3—241no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—21no—
308C32×C4—21no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—32no—
312C32×C4—41no—
313C32×C4—41no—
314C32×C4—42no—
315C32×C4—42no—
316C32×C4—42no—
317C32×C4—44no—
318C32×C4—41no—
319C32×C4—41no—
320C32×C4—41no—
321C32×C4—61no—
322C32×C4—121no—
323C32×C4—121no—
324C32×C4#23511yescharacteristic
325C32×C4D3×D411yescharacteristic
326C32×C4D3×D411yescharacteristic
327C32×C4#23711yescharacteristic
328C32×C4C2×C3×D411yescharacteristic
329C32×C4C3×#4111yescharacteristic
330C3×Dic3—22no—
331C3×Dic3—21no—
332C3×Dic3—21no—
333C3×Dic3—22no—
334C3×Dic3—42no—
335C3×Dic3—41no—
336C3×Dic3—41no—
337C3×Dic3—41no—
338C3×Dic3—41no—
339C3×Dic3—81no—
340C3×Dic3#21512yescomplemented
341C3×Dic3C3×SD1612yescomplemented
342C2×SD16—91no—
343C2×SD16—91no—
344C4⋊Q8—31no—
345C4×D4—91no—
346C4⋊C8—91no—
347#103—31no—
348#103—91no—
349C33#46111yescharacteristic, complemented, Sylow
350C23×C3—181no—
351C22×D3—61no—
352C22×D3—61no—
353C22×D3—121no—
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—21no—
357C3×Q8—21no—
358C3×Q8—21no—
359C3×Q8—21no—
360C3×Q8—21no—
361C3×Q8—21no—
362C3×Q8—41no—
363C3×Q8—41no—
364C3×Q8—41no—
365C3×Q8—41no—
366C3×Q8—42no—
367C3×Q8—42no—
368C3×Q8—61no—
369C3×Q8—61no—
370C3×Q8—61no—
371C3×Q8—61no—
372C3×Q8—121no—
373C3×Q8—121no—
374C3×D4—92no—
375C3×D4—181no—
376C2×C3×C4—22no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—31no—
384C2×C3×C4—31no—
385C2×C3×C4—31no—
386C2×C3×C4—31no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—42no—
390C2×C3×C4—41no—
391C2×C3×C4—61no—
392C2×C3×C4—61no—
393C2×C3×C4—62no—
394C2×C3×C4—61no—
395C2×C3×C4—61no—
396C2×C3×C4—61no—
397C2×C3×C4—61no—
398C2×C3×C4—121no—
399C2×C3×C4—362no—
400C2×C3×C4C2×D3211yescharacteristic
401C2×C3×C4C22×C3×D311yescharacteristic
402C2×C3×C4C22×C3×D311yescharacteristic
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3C3×#6711yescharacteristic
406C2×Dic3C3×#6711yescharacteristic
407D12—32no—
408D12—32no—
409D12—62no—
410D12—61no—
411D12—61no—
412D12—121no—
413C4×D3—122no—
414C4×D3—122no—
415Dic6—21no—
416Dic6—21no—
417C3×C8—61no—
418C3×C8—61no—
419C3×C8—121no—
420C3×C8—181no—
421C3⋊C8—21no—
422C3⋊C8—62no—
423C3⋊C8—61no—
424C3⋊C8—61no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—21no—
434C2×C32—41no—
435C2×C32—41no—
436C2×C32—41no—
437C2×C32#88411yescharacteristic
438C2×C32D3×SD1611yescharacteristic
439C2×C32#91811yescharacteristic
440C2×C32#93811yescharacteristic
441C2×C32C2×#21511yescharacteristic
442C2×C32#94211yescharacteristic
443C2×C32C3×#12111yescharacteristic
444C2×C32C2×C3×SD1611yescharacteristic
445C2×C32C3×#13711yescharacteristic
446C3⋊D3—42no—
447C3×D3—122no—
448C3×D3—122no—
449C3×D3—242no—
450C2×Q8—31no—
451C2×Q8C3×D3211yescharacteristic
452C2×D4—91no—
453C22×C4—181no—
454SD16—182no—
455SD16—182no—
456C2×C8—31no—
457C2×C8—91no—
458C4⋊C4—31no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C22⋊C4—181no—
462C42—31no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—41no—
467C22×C3—182no—
468C22×C3—361no—
469C22×C3—361no—
470C22×C3D3×#6711yescharacteristic
471C22×C3C3×D3×D411yescharacteristic
472C22×C3C2×C3×#6711yescharacteristic
473C2×D3—62no—
474C2×D3—62no—
475C2×D3—122no—
476C2×D3—121no—
477C2×D3—121no—
478C2×D3—121no—
479C2×D3—121no—
480C2×D3—241no—
481C2×D3—241no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—21no—
485C3×C4—21no—
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—32no—
489C3×C4—32no—
490C3×C4—42no—
491C3×C4—42no—
492C3×C4—42no—
493C3×C4—44no—
494C3×C4—41no—
495C3×C4—41no—
496C3×C4—41no—
497C3×C4—41no—
498C3×C4—41no—
499C3×C4—62no—
500C3×C4—62no—
501C3×C4—61no—
502C3×C4—61no—
503C3×C4—122no—
504C3×C4—121no—
505C3×C4—121no—
506C3×C4—121no—
507C3×C4—121no—
508C3×C4—241no—
509C3×C4#380411yescharacteristic
510C3×C4#381111yescharacteristic
511C3×C4C3×#23511yescharacteristic
512C3×C4C3×D3×D411yescharacteristic
513C3×C4C3×D3×D411yescharacteristic
514C3×C4C3×#23711yescharacteristic
515Dic3—21no—
516Dic3—41no—
517Dic3—41no—
518Dic3C3×#21512yescomplemented
519C32—21no—
520C32—21no—
521C32—21no—
522C32—41no—
523C32#481811yescharacteristic, complemented
524C32#508311yescharacteristic, complemented
525C32C3×#46111yescharacteristic, complemented
526C23—181no—
527Q8—21no—
528Q8—21no—
529Q8—61no—
530Q8—61no—
531D4—92no—
532D4—181no—
533C2×C4—21no—
534C2×C4—31no—
535C2×C4—31no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—362no—
539C2×C4C2×C3×D3211yescharacteristic
540C8—61no—
541C8—181no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—41no—
552C2×C3—41no—
553C2×C3—41no—
554C2×C3—362no—
555C2×C3D3×#21511yescharacteristic
556C2×C3#6394811yescharacteristic
557C2×C3#6395811yescharacteristic
558C2×C3C3×#88411yescharacteristic
559C2×C3C3×D3×SD1611yescharacteristic
560C2×C3C3×#91811yescharacteristic
561C2×C3C3×#93811yescharacteristic
562C2×C3C2×C3×#21511yescharacteristic
563C2×C3C3×#94211yescharacteristic
564D3—122no—
565D3—122no—
566D3—242no—
567C22—182no—
568C22—361no—
569C22—361no—
570C22C3×D3×#6711yescharacteristic
571C4—32no—
572C4—41no—
573C4—61no—
574C4—121no—
575C4—121no—
576C4C3×#380411yescharacteristic
577C4C3×#381111yescharacteristic
578C3—21no—
579C3—21no—
580C3—21no—
581C3—41no—
582C3#1059156511yescharacteristic, direct product factor
583C3C3×#481811yescharacteristic, complemented
584C3C3×#508311yescharacteristic, complemented
585C2—362no—
586C2C3×D3×#21511yescharacteristic
587C2C3×#6394811yescharacteristic
588C2C3×#6395811yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC2×C3×Q8D32
hypocenterC32C3×#461
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C32×#944C2semidirect
5. C32×#835C2semidirect
6. C3×Dic3⋊(C3⋊C8)C2semidirect
7. C3×#63565C2semidirect
10. #10591565C3direct (N × H)
27. C3×#944D3semidirect
28. C3×#944C2×C3semidirect
41. C3×#835D3semidirect
42. C3×#835C2×C3semidirect
46. Dic3⋊(C3⋊C8)C2×C3semidirect
47. #63565C2×C3semidirect
129. C32×Dic3SD16semidirect
136. #944C3×D3semidirect
150. #835C3×D3semidirect
340. C3×Dic3#215semidirect
341. C3×Dic3C3×SD16semidirect
349. C33#461semidirect
518. Dic3C3×#215semidirect
523. C32#4818semidirect
524. C32#5083semidirect
525. C32C3×#461semidirect
582. C3#10591565direct (N × H)
583. C3C3×#4818semidirect
584. C3C3×#5083semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×#37552
C32×#9442
C3×#638942
C2×C3×#37172
C32×#8352
C3×Dic3⋊(C3⋊C8)2
C3×#635652
C3×#50833
C3×#48183
#105915653